Overlap Number of Graphs Daniel W. Cranston Virginia Commonwealth University dcranston@vcu.edu Slides available on my preprint page Joint with Nitish Korula, Tim LeSaulnier, Kevin Milans Chris Stocker, Jenn Vandenbussche, and Doug West Atlanta Lecture Series V 26 February 2012
Definitions Def: A set overlaps another set if they intersect but neither contains the other.
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap.
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f .
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 126 134 13
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 126 234 13
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 126 234 13 so ϕ ( G ) ≤ 7
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 26 45 126 234 12 14 13 13 so ϕ ( G ) ≤ 7
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 26 45 126 234 12 14 13 13 so ϕ ( G ) ≤ 7 so ϕ ( G ) ≤ 6
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 26 45 1345 45 126 234 12 14 12 14 13 13 13 so ϕ ( G ) ≤ 7 so ϕ ( G ) ≤ 6
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 26 45 1345 45 126 234 12 14 12 14 13 13 13 so ϕ ( G ) ≤ 7 so ϕ ( G ) ≤ 6 so ϕ ( G ) ≤ 5
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 26 45 1345 45 126 234 12 14 12 14 13 13 13 so ϕ ( G ) ≤ 7 so ϕ ( G ) ≤ 6 so ϕ ( G ) ≤ 5 Def: A pure overlap representation f of a graph G is an overlap representation where no set contains another. The pure overlap number Φ ( G ) is the minimum size of f .
Definitions Def: A set overlaps another set if they intersect but neither contains the other. An overlap representation f of a graph G assigns sets to V ( G ) so that uv ∈ E ( G ) iff f ( u ) and f ( v ) overlap. The overlap number ϕ ( G ) is the minimum size of f . 67 45 26 45 1345 45 126 234 12 14 12 14 13 13 13 so ϕ ( G ) ≤ 7 so ϕ ( G ) ≤ 6 so ϕ ( G ) ≤ 5 Def: A pure overlap representation f of a graph G is an overlap representation where no set contains another. The pure overlap number Φ ( G ) is the minimum size of f . So ϕ ( G ) ≤ 5, but Φ ( G ) ≤ 6.
Main Results Thm 1: We have a linear-time algorithm to determine ϕ ( T ) for every tree T . Corollary: ϕ ( T ) ≤ | T | .
Main Results Thm 1: We have a linear-time algorithm to determine ϕ ( T ) for every tree T . Corollary: ϕ ( T ) ≤ | T | . Thm 2: If G is a planar n -vertex graph and n ≥ 5, then ϕ ( G ) ≤ 2 n − 5, which is sharp for n = 8 and n ≥ 10.
Main Results Thm 1: We have a linear-time algorithm to determine ϕ ( T ) for every tree T . Corollary: ϕ ( T ) ≤ | T | . Thm 2: If G is a planar n -vertex graph and n ≥ 5, then ϕ ( G ) ≤ 2 n − 5, which is sharp for n = 8 and n ≥ 10. Thm 3: If G is an arbitrary n -vertex graph and n ≥ 14, then ϕ ( G ) ≤ n 2 / 4 − n / 2 − 1, which is sharp for even n .
Preliminaries Decomposition Bound: Let F be a decomposition of graph G into cliques of order at most k , where k ≥ 2. If δ ( G ) ≥ k , then Φ ( G ) ≤ |F| . In particular, δ ( G ) ≥ 2 implies Φ ( G ) ≤ | E ( G ) | .
Preliminaries Decomposition Bound: Let F be a decomposition of graph G into cliques of order at most k , where k ≥ 2. If δ ( G ) ≥ k , then Φ ( G ) ≤ |F| . In particular, δ ( G ) ≥ 2 implies Φ ( G ) ≤ | E ( G ) | . Pf: Give each clique in F its own label, and give each vertex all the labels of cliques that contain it.
Preliminaries Decomposition Bound: Let F be a decomposition of graph G into cliques of order at most k , where k ≥ 2. If δ ( G ) ≥ k , then Φ ( G ) ≤ |F| . In particular, δ ( G ) ≥ 2 implies Φ ( G ) ≤ | E ( G ) | . Pf: Give each clique in F its own label, and give each vertex all the labels of cliques that contain it. Prop: If G is triangle-free, then Φ ( G ) ≥ | E ( G ) | , and Φ ( G ) = | E ( G ) | when δ ( G ) ≥ 2 .
Preliminaries Decomposition Bound: Let F be a decomposition of graph G into cliques of order at most k , where k ≥ 2. If δ ( G ) ≥ k , then Φ ( G ) ≤ |F| . In particular, δ ( G ) ≥ 2 implies Φ ( G ) ≤ | E ( G ) | . Pf: Give each clique in F its own label, and give each vertex all the labels of cliques that contain it. Prop: If G is triangle-free, then Φ ( G ) ≥ | E ( G ) | , and Φ ( G ) = | E ( G ) | when δ ( G ) ≥ 2 . Pf: We can’t do better than one label on each edge.
Preliminaries Decomposition Bound: Let F be a decomposition of graph G into cliques of order at most k , where k ≥ 2. If δ ( G ) ≥ k , then Φ ( G ) ≤ |F| . In particular, δ ( G ) ≥ 2 implies Φ ( G ) ≤ | E ( G ) | . Pf: Give each clique in F its own label, and give each vertex all the labels of cliques that contain it. Prop: If G is triangle-free, then Φ ( G ) ≥ | E ( G ) | , and Φ ( G ) = | E ( G ) | when δ ( G ) ≥ 2 . Pf: We can’t do better than one label on each edge. Deletion Bound: If v is a vertex with d ( v ) ≤ 2 in a graph G with at least 3 vertices, then Φ ( G ) ≤ Φ ( G − v ) + 2. If d ( v ) ≤ 1, then ϕ ( G ) ≤ ϕ ( G − v ) + 2.
Preliminaries Decomposition Bound: Let F be a decomposition of graph G into cliques of order at most k , where k ≥ 2. If δ ( G ) ≥ k , then Φ ( G ) ≤ |F| . In particular, δ ( G ) ≥ 2 implies Φ ( G ) ≤ | E ( G ) | . Pf: Give each clique in F its own label, and give each vertex all the labels of cliques that contain it. Prop: If G is triangle-free, then Φ ( G ) ≥ | E ( G ) | , and Φ ( G ) = | E ( G ) | when δ ( G ) ≥ 2 . Pf: We can’t do better than one label on each edge. Deletion Bound: If v is a vertex with d ( v ) ≤ 2 in a graph G with at least 3 vertices, then Φ ( G ) ≤ Φ ( G − v ) + 2. If d ( v ) ≤ 1, then ϕ ( G ) ≤ ϕ ( G − v ) + 2. Pf: Easy for Φ , and not too hard for ϕ .
Preliminaries (part 2) Edge Bound: If δ ( G ) ≥ 2 and G � = K 3 , then ϕ ( G ) ≤ | E ( G ) | − 1.
Preliminaries (part 2) Edge Bound: If δ ( G ) ≥ 2 and G � = K 3 , then ϕ ( G ) ≤ | E ( G ) | − 1. Pf: Slightly modify a pure overlap labeling of size | E ( G ) | .
Preliminaries (part 2) Edge Bound: If δ ( G ) ≥ 2 and G � = K 3 , then ϕ ( G ) ≤ | E ( G ) | − 1. Pf: Slightly modify a pure overlap labeling of size | E ( G ) | . Def: A star-cutset in a graph G is a separating set S containing a vertex x adjacent to all of S − x .
Preliminaries (part 2) Edge Bound: If δ ( G ) ≥ 2 and G � = K 3 , then ϕ ( G ) ≤ | E ( G ) | − 1. Pf: Slightly modify a pure overlap labeling of size | E ( G ) | . Def: A star-cutset in a graph G is a separating set S containing a vertex x adjacent to all of S − x . Edge Lower Bound: If G is a triangle-free graph with no star-cutset, then ϕ ( G ) ≥ | E ( G ) | − 1.
Preliminaries (part 2) Edge Bound: If δ ( G ) ≥ 2 and G � = K 3 , then ϕ ( G ) ≤ | E ( G ) | − 1. Pf: Slightly modify a pure overlap labeling of size | E ( G ) | . Def: A star-cutset in a graph G is a separating set S containing a vertex x adjacent to all of S − x . Edge Lower Bound: If G is a triangle-free graph with no star-cutset, then ϕ ( G ) ≥ | E ( G ) | − 1. Pf idea: We can’t do anything better than in the Edge Bound.
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